\section{Fractionation}
\label{appendix:fractionation}

Although the mixing in the early solar system was very efficient, the
compositions of different objects nowadays are quite different. This is 
because most of the objects in our solar system have experienced various
fractionation, both elementally and isotopically, to different extent.

%----------------------------------------------------------------------------
\subsection{Elemental Fractionation}\label{sect:elemFrac}
%----------------------------------------------------------------------------

The Earth's atmosphere is
an example of element fractionation.  It is abundant in nitrogen and oxygen
(in the form of ${\rm N}_2$ and various oxygen molecules such as ${\rm O}_2$
and ozone) and significantly depleted in hydrogen and helium.
This is because the gravity of Earth is strong enough to retain the former 
species but any ${\rm H}_2$ or He introduced into the Earth's atmosphere
leaks out into space on geological time scales.  Another example is the
Earth's core/mantle separation.  Iron and nickel, due to their high density,
preferentially sank to the core during the Earth's molten phase while
silicates preferentially floated to the surface.  The Earth thus has
a Fe/Ni core and a silicon rich mantle.  Chemistry and gravity combine to
separate the elements strongly in planets.

%----------------------------------------------------------------------------
\subsection{Isotopic Fractionation}\label{sect:isotopicFrac}
%----------------------------------------------------------------------------

By contrast, isotopic fractionation is generally small in planetary samples.
There are many processes that cause isotopic fractionation, and they are
generally characterized as {\em mass dependent} or {\em mass independent}.
An example of a mass-dependent process is water vapor condensation in a
rain cloud.  It is an experimental rule of thumb that, in equilibrium,
the denser the phase of a material, the more it tends to be enriched in the
heavier isotopes of a constituent element.  Thus, in equilibrium, ice
will tend to be enriched in $^{18}$O compared to liquid water and liquid
water will be enriched in $^{18}$O compared to vapor.  Thus, a rain drop
will tend to be rich in $^{18}$O relative to the vapor it condenses from.

Another mass-dependent fractionation process is kinetic fractionation, which
occurs when chemical reactions occur out of equilibrium, especially when
the reaction products become physically separated from the reactants.
For example, if a rock melts, the lighter isotopes of silicon might escape
(as silicate molecules) preferentially because they are lighter and move
faster than silicates containing heavier silicon atoms.

An example of mass-independent fractionation is self-shielding of CO.
If UV radiation impinges on a cloud of CO, the cloud will be thicker to
C$^{16}$O than to C$^{17}$O or C$^{18}$O because the $^{16}$O is so much
more abundant (that the other two isotopes.  Deep in the core of the cloud,
then, the UV intensity at the
frequencies that can dissociate C$^{16}$O are much reduced compared to
those than can dissociate C$^{17}$O or C$^{18}$O.  We could thus expect
$^{17}$O and $^{18}$O to preferably be in atomic form deep in the cloud
core.  If some other chemical process locks up the oxygen into, say, ice,
that ice will be preferentially enriched in the heavy isotopes of oxygen.
This process has been proposed as the source of the 5\% enrichment in
the heavy isotopes of oxygen in Earth's ocean water compared to the inferred
isotopic composition of the Sun.

We know from the presolar grains recovered from primitive meteorites that
much of the ejecta from these stars were highly anomalous in their isotopic
composition compared to the Solar System average composition.

%----------------------------------------------------------------------------
\subsection{Oxygen}\label{sect:oxygen}
%----------------------------------------------------------------------------

Oxygen isotope abundance variations in meteorites are very useful in 
elucidating chemical and physical processes that occurred during the 
formation of the solar system \cite{1993AREPS..21..115C}.

Fig.\ref{fig:oxygen} shows the oxygen three isotope plot.
The $\delta^{18}$O notation is defined as (like Eq. \ref{eq:delta},
same for $^{17}$O):

\begin{equation}
\delta \oxygen{18} =
  \left[
    \frac{ (\oxygen{18}/\oxygen{16})_{sample} }
         { (\oxygen{18}/\oxygen{16})_{standard} }
    - 1
  \right] \times 1000
\label{eq:oxygen}
\end{equation}

The SMOW stands for Standard Mean Ocean Water.
The CAIs lie on the slope 1 line, while the Earth and Moon lie on the slope 0.5
line and they are
$^{17,18}$O-rich compared to the
CAIs and apparently the Sun, which lies at the left bottom conner.
The evolution path of oxygen isotopes seem to be like this: it starts from
the left bottom conner and spreads along the slope 1 line due to 
self-shielding (mass-independent). 
(see Sec. \ref{sect:isotopicFrac})
The different positions of CAIs along the line indicate their
formation sites in the solar nebula. For the terrestrial objects they share
the similar initial oxygen isotope ratio and spread along the slope 0.5 line
due to mass-dependent fractionation.
See Sec. \ref{sect:derivation} for a rough estimation.
It's just like Robert Clayton's words, ``the Sun is not anomalous, you are!''

CAIs seem to be condensates from the solar gas and are
$^{16}$O-rich. Also the FUN CAIs do not seem to be much different
in their oxygen from the ``regular CAIs'', although they lie on a slope 0.5
fractionation line with regular CAIs.
%Explain in Dimitri Papanastassiou's words that the FUN CAIs are ``not oddballs''. 

\begin{figure}[ht!]
\centering
\includegraphics[width=\figuresize\textwidth]{figures/oxygen}
\caption{
Oxygen three-isotope plot showing representative compositions of 
major primary components of solar system matter, the solar wind (SW), 
and our preferred value for the Sun. All data fall predominantly on a 
single mixing line characterized by excesses (lower left) or depletions 
(upper right) of 16O relative to all samples of the Earth and Moon. 
Plotted are the most 16O-enriched solar system samples: an unusual 
chondrule (47); individual platy hibonite grains (55), which are 
ultra-refractory oxides from carbonaceous chondrites (CC); water 
inferred to have oxidized metal to magnetite (56) in ordinary 
chondrites (OC); very 16O-depleted water from the CC Acfer 094 (3), 
and whole CAIs from CC (19); and chondrules from CC and OC (19), 
bulk Earth (mantle), and Mars (SNC meteorites). The mass-dependent 
fractionation trajectory of primary minerals in FUN inclusions and 
the pure 16O (slope 1.0) line (57) are also shown. 
}
\label{fig:oxygen}
\end{figure}

Figure and caption from \cite{2011Sci...332.1528M}

%-------------------------------------------------------------------------------
\subsection{Derivation of Slope 0.5 Line}\label{sect:derivation}
%-------------------------------------------------------------------------------

To simply illustrate the idea of fractionation, we consider an oxygen atom
at one end of a spring (in molecules). The oscillation energy is then 
proportional to the frequency $\omega$:
%
\begin{equation}
E \propto \omega \sim \sqrt{ \frac{1}{m} }
\end{equation}
%
The last step has applied the harmonic oscillation approximation. 

For oxygen there are three stable isotopes: $^{16}O$, $^{17}O$, $^{18}O$. So

\[
\sqrt{ \frac{1}{m} } =
  \sqrt{ \frac{1}{m_{16}+\Delta m} } =
  \sqrt{ \frac{1}{m_{16}} } \left( 1 + \frac{\Delta m}{m_{16}} \right)^{-1/2} =
  \sqrt{ \frac{1}{m_{16}} } 
    \left( 1 - \frac {\Delta m} {2m_{16}} \right)
\]

where $\Delta m=0$ for $^{16}O$, 1 for $^{17}O$ and 2 for $^{18}O$, in the unit
of atomic mass unit (roughly). The last step is for $\Delta m \ll m_{16}$.
For energy we have:

\[
E = E_{16} \left( 1 - \frac {\Delta m} {2m_{16}} \right)
\] 

For Maxwell-Boltzmann distribution, we can write the number ratio of $^{17}O$
to $^{16}O$ as:

\[
\frac {^{17}O/\oxygen{17}_{standard}}{^{16}O/\oxygen{16}_{standard}} = 
  \frac { e^{ -E_{17}/kT } } { e^{ -E_{16}/kT } } =
  exp\left[ \frac{E_{16}} {kT} \frac {\Delta m} {2m_{16}} \right] =
  1 + \frac{E_{16}} {kT} \frac {\Delta m} {2m_{16}}
\]

In the last step the oscillation energy could be considered similar to $kT$ and
$\Delta m \ll m_{16}$ is applied.

The $\delta$ notation is usually defined as:

\[
\delta^{17}O = 
  \left[ \frac { (^{17}O/^{16}O)_{sample} } { (^{17}O/^{16}O)_{standard} }
         - 1
  \right] 
  \times 1000 
\]

So for two samples, we have

\[
\delta^{17}O_2 - \delta^{17}O_1 = 
  \frac{ (^{17}O/^{16}O)_2 - (^{17}O/^{16}O)_1 }{ (^{17}O/^{16}O)_{standard} }  
    \times 1000 =
  \frac{ \Delta m }{ 2m_{16} } 
    \left[ 
      \left( \frac{E_{16}}{kT} \right)_2 - \left( \frac{E_{16}}{kT} \right)_1
    \right]
    \times 1000
\]

\[
\frac{ \delta^{17}O_2 - \delta^{17}O_1 } 
  { \delta^{18}O_2 - \delta^{18}O_1 } =
  \frac{ \Delta m_{17} }{ \Delta m_{18} } =
  \frac 12
\]

